Mayer-Cluster Expansion of Instanton Partition Functions and Thermodynamic Bethe Ansatz

Abstract

In arXiv:0908.4052, Nekrasov and Shatashvili pointed out that the N=2 instanton partition function in a special limit of the Omega-deformation parameters is characterized by certain thermodynamic Bethe ansatz (TBA) like equations. In this work we present an explicit derivation of this fact as well as generalizations to quiver gauge theories. To do so we combine various techniques like the iterated Mayer expansion, the method of expansion by regions, and the path integral tricks for non-perturbative summation. The TBA equations derived entirely within gauge theory have been proposed to encode the spectrum of a large class of quantum integrable systems. We hope that the derivation presented in this paper elucidates further this completely new point of view on the origin, as well as on the structure, of TBA equations in integrable models.

v2: typos corrected and references added. 27+15 pages, 4 figures; v3: preprint number added

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• A slow review of the AGT correspondence
• A Solvable Deformation of Quantum Mechanics
• Blowup Equations for Refined Topological Strings
• Exact Quantization Conditions, Toric Calabi-Yau and Nonperturbative Topological String
• Comments on Exact Quantization Conditions and Non-Perturbative Topological Strings
• Classical limit of irregular blocks and Mathieu functions
• Spherical Hecke algebra in the Nekrasov-Shatashvili limit
• Finite ${\epsilon }_2$-corrections to the $\mathcal{N}=2$ SYM prepotential
• Fermions and scalars in $\mathcal{N}=4$ Wilson loops at strong coupling and beyond
• Strong Wilson polygons from the lodge of free and bound mesons
• Mayer expansion of the Nekrasov pre potential: the subleading ${\epsilon }_2$-order
• Classical Liouville Three-point Functions from Riemann-Hilbert Analysis
• Quantum integrability of $\mathcal{N}=2$ 4d gauge theories
• Seiberg-Witten period relations in Omega background
• Quantum Periods and TBA-like Equations for a Class of Calabi-Yau Geometries
• Solution of quantum integrable systems from quiver gauge theories
• Quantum Analytic Langlands Correspondence